Differential analyzer



Aug. 15, 1950 c. A. LOVELL DIFFERENTIAL ANALYZER 2 Sheets-Sheet 1 FiledSept. 9, 1948 FIG.

wv- TOR C. A. LOVE LL a. H. )MM- ATT ORNEV Aug. 15, 1950 c,A,-LOVE| L 7.2,519,262

' D iFFERE NTIAL ANALYZER Filed Sept. 9, 1948 2 Sheets-Sheet? FIG. 2

' wws/vrop C. A. LOVEL L ilk-W di.

Patented Aug. 15 1950 UNlTED STATES ATENT OFFICE DIFFERENTIAL ANALYZERApplication September 9, 1948, Serial No. 48,358

-6 Claims.

This invention relates to an improvement in electrical differentialanalyzers, making possible the solution, by electrical means inconjunction with servomotors, of simultaneous differential equations andof a differential equation in which the unknown quantity to be computedis complex and the terms of the equation include complex coeflicients.

The present invention is an improvement over that of my copendingapplication Serial No. 440,504, Electrical Computing System, filed April25, 1942, now Patent No. 2,476,747. That application describes anelectromechanical system for solving simultaneous linear algebraicequations and is assigned to the same assignee as the present invention.

It is the object of the invention herein disclosed to provide a systemof apparatus for automatically solving simultaneous differentialequations of any degree in p, where p is the differential operator Thecircuits shown in the figure are linear and of the first degree in p.Higher degrees in p are solved through successive differentiation usingdifierentiators of the type shown. Coeflicients which are non-linearfunctions of the variables may be included through use of taperedpotentiometers in place of the linear potentiometers shown.

The system of apparatus herein disclosed may with slight modification beadapted to the solution of a differential equation, of any degree in p,of a single complex variable when the coefficients and the constant termare themselves complex. Such an equation may be restated in the form oftwo simultaneous equations each involving the real and imaginary parts,and their time derivatives, of the unknown quantity and the like partsof the constant term and of the coeificients.

A further object of the invention is therefore to provide a system ofapparatus for the automatic solution of a differential equation of thefirst degree in p, in which the unknown quantity, the coefiicients andthe constant term are complex.

The first time derivative of a variable V may be written either pV or V.In convenience, the third form will herein be used.

Electrical difierentiators are used in the apparatus of the invention.These provide derivatives with respect to time, for which reason theindependent variables must be a function of time or be transformed to besuch in any wellknown manner. The apparatus then furnishes continuoussolutions of simultaneous differential equations in which theindependent variable is a function of time, and to provide a novel apparatus for obtaining such continuous solutions is also an object of theinvention.

In what follows, the variables involved in the equation to be solvedwill be considered, either inherently or by suitable transformation,functions of time.

The invention will be understood from the following description,referring to the accompanying drawings, in which:

Fig. 1 is a diagram of a circuit for the solution of two simultaneousequations of the form Fig. 2 is a diagram of a circuit for the solutionof an equation of the form In both figures, like numerals and lettersindicate like elements.

It is convenient to mention the sources of various known devices whichsymbolically are shown in the figures of the drawings. Reversingamplifiers, identified by the symbol RA, are well known; they includeeach a thermionic vacuum tube in a conventional circuit of unit gainproviding an output voltage equal but opposite in sign to an inputvoltage. Summing amplifiers, 2A, are those disclosed by K. D. Swartzel,Jr., in United States Patent 2,401,779, June 11, 1946. Electricaldifferentiators, D, are the summing amplifiers of Swartzel so modifiedthat the out= put voltage is the negative first time derivative of thesum of the input voltages; such modification is disclosed in UnitedStates Patent 2,412,227, December 10, 1946, to H. G. Och and K. D.Swartzel, Jr. A ternatively to the differenitators, D, there may be usedthe diiferentiating circuit shown in United States Patent 2,251,- 973,August 12, 1941, to E. S. L. Beale et a1. Many types of servomotors, S,are known; directcurrent servomotor circuits are shown, for example, inUnited States Patents 1,086,729, February 10, 1914, to J. A. Rey, and1,268,712, June 4, 1918, to F. A. H. Harl. P. A. Borden, in.

United States Patent 2,114,330, April 19, 1938, discloses the shaping ofpotentiometer cards to make the brush voltage any desired function ofthe brush displacement from a reference position.

Referring now to Fig. 1, it is convenient to write the equations to besolved in the equivalent forms In the circuit of Fig. 1, summingamplifier I controls servomotor II, positioning brush I2 onpotentiometer I3 which shunts battery I4, the mid-point thereof beinggrounded, ultimately deriving at brush I2 a voltage proportional to x.Similarly, summing amplifier IIO drives servomotor I II to cause brush II2 ultimately to derive a voltage proportional to y from potentiometerII3 shunting battery I I 4.

Batteries I4 and I I4, of which the mid-points are grounded, areshunted, respectively, by potentiometers I5 and I I5. The positive andnegative voltages of battery I4 are chosen at least equal to the maximumnumerical value of the voltages of battery H4 similarly correspond tothe maximum numerical value of Voltages proportional to are selected,respectively, by handset taps I6 and H6.

The voltage at brush I2 is difierentiated with respect to time bydifferentiator IT to provide across potentiometer IS the voltage -a:when that at brush I2 is .12. Similarly, through differentiator III, thevoltage y is provided across potentiometer H9 when that at brush H2 isy. These time derivative voltages are fractionated, by handset taps 2Dand I20, respectively, proportionally to Reversing amplifiers I8 and IE8provide voltages numerically equal but opposite in sign to the voltagesat brushes I2 and H2, respectively.

Writing :12 and y for the instantaneous values of voltage at brushes I2and H2, respectively, one sees that the input voltages at amplifier II)are: a:' through resistor 25,

and

from tap I6 through resistor 26, and

through resistor .27 from tap I20. To amplifier I ID the input voltagesare: y through resistor I25,

from tap HG through resistor I26, and

from tap through resistor I21.

Servomotors II and III may be of the form shown by S. Darlington inUnited States Patent 2,438,112, March 23, 1948, assigned to the sameassignee as the present invention. Summing ampiifiers I2 and III] arestabilized by reverse feedback through resistors 28 and I28,respectively. Potentiometers I 3, I5 and H3, H5 are grounded each at itsmid-point, as are batteries I4, I4 and H4, H4. Thus motors I! and IIIare driven, when set in operation by suitable switches (not shown), toposition their corresponding brushes I2 and H2 to either positive ornegative voltages as may be required for the simultaneous adjustment tozero of the total input voltages,

to amplifier I I0. When this adjustment is completed, the voltages atbrushes I2 and I22 are respectively proportional to :1: and y. Scalesconcentric with potentiometers I3 and I I3 may be provided on which thecomputed values of a: and 3/ may be read.

Fig. 2 shows the circuit of Fig. 1 modified for the solution of theequation (a+jb)z+(c+jd)z+ (m-H'n) =0 where z=a+j 8, and a and p are tobe found.

Expanding and separately collecting the real and the imaginary terms ofthe equation to be 7 solved, we obtain two simultaneous equations in a.and ii and their first time derivatives, thus transforming the singleequation in a complex variable with complex coefficients and a complexconstant term into two simultaneous equations in which the real andimaginary parts of the complex variable take the places of x and y inthe circuit of Fig. 1. The two simultaneous equations so obtained are:

25 fl+ fi+ In Fig. 2, servomotor I 30, controlled by the output ofsumming amplifier I3I, operates to find a, while servomotor I6!) iscontrolled by summing amplifier I6I to find ,8. The constant terms m andn are introduced to the respective summing amplifiers I3I and I6I overresistors I32 and 62 from batteries I33 and I63, respectively. Motor I36drives shaft I34 carrying brush I35 on potentiometer I36 and brush I3?on potentiometer I38, each of which potentiometers is grounded at itsmid-point. Across potentiometer I36 is applied a voltage a, derived athandset tap I39 on potentiometer I shunting battery I4I, while a voltage60 b is applied across potentiometer I38 from handset tap I42 onpotentiometer I43 shunting battery I44. Assuming, as is ultimately thecase, that the angular positions of brushes I35 and I3? from thegrounded mid-points represent a, the voltage at brush 535 is (la andthat at brush I3"! is b6.

Similarly, motor I drives shaft I64 ultimately to position brushes I65and. I6? at the angle ,8 from the grounded mid-points of potentiometersI66 and I68. To potentiometer I 65 is supplied 70 the voltage a from tapI39; to potentiometer I68 the voltage -b is applied from handset tap I92on potentiometer I93 shunting battery I94. The voltages at brushes I65,I 61 are then afi, -b,3, respectively. The voltage (la from brush I35 ispassed on con.

duotor 145 through resistor I43 to the input of amplifier ISI; to thesame input through reversing amplifier I47, condenser I48 and feed-.back amplifier I49, the voltage Cu is passed through resistor II. Theadjustable feedback resistor I controls the gain of amplifier I49 and isset to provide the factor 0. Through resistors I52 and I53, amplifier[3| receives the voltages be on line I69 from brush I61, and -dfi frombrush I through condenser Ill) and amplifier III of which feedbackresistor I12 is varied to provide he f or d- The nput voltage o amp t!are then as follows:

Through lta e Resistor To summing amplifier I6I, the voltage as is feddirectly from brush I65 through resistor I15, and the voltage bu onconductor I'I6 from brush I3! is fed through resistor Ill. The lattervoltage is reversed by reversing amplifier I18 and passed throughcondenser I19 to amplifier I with adjustable feedback resistor IBI. Theoutput of amplifier I80 is the voltage (1a, reaching amplifier I6Ithrough resistor I82. The remaining input voltage, cc, required foramplifier I6I is derived from the voltage bc on line I69 bydifferentiation and factor adjustment through condenser I55 andamplifier I56 with variable feedback resistor I51; this derivativevoltage reaches amplifier I6I through resistor I83. The

Stabilizing feedback is provided through resistor I to amplifier I3I andthrough resistor I06 to amplifier I6I,

The voltages controlling servomotors I30 and I60. are the negatives ofthe sums of the component input voltages to amplifiers I3I and I6I, andthe connections to motors I30 and I60 are made such that shafts I34 andI64 are driven to angular positions a and [3 when the voltage s ms e eprately an simultan s y br ght to zero, at least to within the smallerror corresponding to the motor power required to overcome brush andbearing friction. This error may be reduced substantially to zero byproper choice of motors, amplifier gains and potentiometer structures, amatter with which the present invention is not concerned. Scales to readthe angular positions of shafts I34 and I64 may be provided to read aand b, respectively.

The coefficients a, b, c and d may, of course, be

is a matter of convenience, however, to keep equal and unchanged theamplifier input resistors, which are required to be large, and toestablish the coefficients by such manipulation as has been described.

The apparatus arrangement for each of the illustrative problems solvedby the circuits of Figs. 1 and 2 is easily made when the analog of aphysical system is to be set up. In such a case it is known in whichdirection each of the variables controlled by the servomotors shouldchange, in response to an error voltage from the correspondingsumming'amplifier, in order to reach the correct solution. The system isthen inherently stable. In the unrestricted case, the stable arrangementof the apparatus is ascertainable, as may be seen from the followingconsiderations:

A differential analyzer of the type of this invention involving 11servos and n unknown variables may be connected in 2 -77.! differentways. A servo may be connected so that a positive error voltage willcause the variable under control to increase or it may be connected sothat the same error sign will cause the variable to decrease. Thischoice of two directions for each of n servos gives 2- polingcombinations. Theoretically at least any one of the n equations may beassigned the duty of controlling the servo which generates any one ofthe n unknown variables. There are n! distinct combinations ofassignments of this sort, and the total number of connections is theproduct of these two factors, which in the problem described is eight.In general the assignments can be made more or less arbitrarily and acombination of polings can be found which causes all the error voltagesto tend toward zero simultaneously. However, some assignments aresuperior to others because a faster convergence toward the solutionresults. The matter of speed of convergence is one best left to beconsidered for each system of equations when it is being set into theapparatus. These remarks apply particularly when the equations are of anabstract nature. The choice of the best assignment is easy when theapparatus is set up as an analog of an existing physical dynamic systemsince the analog will have the dynamic properties of the systemsimulated. The proper assignments are more or less automatic in thiscase.

The dynamic performance of the analyzer components may be made of littleimportance as they affect the over-all system stability, by making theactions of the servos fast compared with the changes in the functionsbeing generated.

In cases where no physical guide to making the assignments and polingconnections is apparent, the criteria for both lie in stabilizingthecoupled system. The dynamical system of the analyzer is represented by asystem of differential equations. An equation, called the characteristicequation of the matrix of such a system, is defined by E. J. Routh inStability of Motion,

' 1877, MacMillan 8; Co., London. Rou'th states a crl'erion forstability which is equivalent to: The motion of a system of bodies isstable if the real roots and the real parts of the complex roots of thecharacteristic equation of the system are all negative. Rouths workapplies to linear systems only. M. Liapounoff in Problerne General de laStabilite du Mouvemerit, Annales de Toulouse, vol. 9, 1907, discussesthe non-linear case. He shows that in the neighborhood of a point thestability conditions for a non-linear system are described precisely bythose of an associated linear system derived from the original system ina prescribed manner. Application of the tests of Routh and Liapounoffare laborious and should be used only in the difficult cases where noreliable physical guides for making the connections are apparent. Routhsand Liapounoifs results are found in classical mathematical treatises,for instance Cours d Analyse Mathematique, vol. 3, Edouard Goursat,Gauthier-Villars, Paris, 1915. It is to be noted that Routh andLiapounoif discuss in part systems found in nature and in part syntheticsystems such as a governed steam engine. Hence, their results applydirectly to the synthetic dynamic systems created by cross-couplingservo-systems in accordance with this invention. The tests show theconditions which are imposed on the properties of the servos andactually guide the user in making the connections. A more complete setof references is found in the 1946 edition of the EncyclopediaBritannica, under Stability.

It should be noted that the solutions of a system of differentialequations are in general variables, and as derived by this invention arevariable functions of time. Some sort of recording device is requiredsuch as automatic plotters or means for photographing output dials atregular time intervals. hence are not shown as parts of the presentinvention.

The solutions of differential equations involve arbitrary constants. Thenumber for each variable is equal to the order of the highest derivativeof that variable which appears in any of the equations. There are,therefore, families of time functions for each variable rather than asingle such function; the respective members of the families beingdistinguished from each other by having different values for theconstants. These constants may be defined by the values the variable andits derivatives have initially, that is at the start of the solutionwhen i=0. If one set of initial conditions are used one time functionresults, another set gives a different member of the family. A completesolution involves a reasonably complete representation of the entirefamily and hence it is desirable to be able to start with any requiredset of initial values. This, however, is not an essential part of theapparatus because it is always possible by a change of variables toreplace any set of equations and associated initial conditions by anequivalent set in which all the initial values are zeros. The solutionsof the original set can then be computed from the solutions of the newset through use of the transformation formulae used in changing thevariables.

Thus the invention described in this disclosure can solve equationshaving any given set of associated initial conditions and it becomesunnecessary to show specific means for setting in initial values for thevariables and their derivatives other than zeros. This means startingthe system from a rest point where all the generated variables are zero.

For the sake of simplicity, the circuit of Fig. 1 has been shown anddescribed as adapted to solve one of the simpler forms of simultaneousdifferential equations in two variables. The additional elements andconnections required to provide a circuit for the solution of a largerplurality of simultaneous differential equations, in an equally largernumber of variables with Such devices are well known and higher orderderivatives and coeflicients which are non-linear functions of certainof the variables, may readily be made and will be understood by oneskilled in the art after reading the foregoing disclosure. It will benoted that each computing channel corresponds to a chosen equation andto a selected one of the variables to be computed and includes aservomotor controlling a variable voltage; the servomotor responds to anerror voltage in the output circuit of a summing amplifier in the inputcircuit of which are algebraically combined voltage representing all ofthe terms of the chosen equation. The several servomotors operatecontemporaneously to vary the voltages they respectively control untilall the error voltages become zero, whereupon the variable voltagesbecome individually proportienal to the variables to be computed.

The automatic solution of a differential equation involving a singlevariable, in which the coefficients may be non-linear, is disclosed andclaimed in the copending application of S. Darlington, filed September10, 1947, Serial No. 775,287, now Patent No. 2,494,036, assigned to thesame assignee as the present invention.

What is claimed is:

1. A system of apparatus for the solution of a plurality of simultaneousdifferential equations involving an equal plurality of variables, eachequation including a known term, a term directly proportional to the onevariable and its time derivatives and proportional to each of the othervariables and their time derivatives, comprising a plurality ofinterconnected computing channels equal in number to the equations andcorresponding individually thereto, each of said channels including afirst and a second source of voltage, means for deriving from the firstsource a first voltage proportional to the known term of thecorresponding equation, controllable means for deriving from the secondsource a variable voltage, means for fractionating the variable voltageproportionally to the term in each equation directly proportional to theone variable, means including electrical diiferentiators for derivingfrom the variable voltage differential voltages respectivelyproportional to the terms of each equation involving the timederivatives of the one variable, a summing amplifier having an input andan output circuit, servomotor means responsive to voltage in the outputcircuit to operate the controllable means and circuit meansinterconnecting the several channels for combining in the input circuitsof the several amplifiers the voltages respectively proportional to theterms of the corresponding equations, thereby producing an error voltagein each of the output circuits which drives the servomotor theretoresponsive so to operate the controllable means that the error voltageis reduced to zero and the variable voltage becomes proportional to theValue of one of the variables satisfying the corresponding equation.

2. Means for the solution of a plurality of simultaneous differentialequations involving a plurality of variables, each of said equationsincluding a known term, terms proportional directly to functions of eachof the variables and terms proportional to the time derivatives ofinvolved orders of each of the variables, comprising a plurality ofchannels equal in number to the equations and corresponding each to onethereof, each of said channels including a source of voltage, means forderiving from the source a voltage proportional and opposite in sign tothe known term of the corresponding equation, a second source ofvoltage, controllable means for deriving from the second source avariable voltage, means for successively differentiating the variablevoltage with respect to time as indicated by the equation, a summingamplifier having an input and an output circuit I110 tor means drivenfrom the output circuit to cperate the controllable means; means inchannel for selecting from the variable voltages fractional voltagesproportional to the terms in each equation involving the severalvariables and for selecting from the differential voltages fractionalvoltages proportional to the terms in each equation involving the timederivatives of the several variables; and interconnecting means forcombining in the input circuit or" each summing amplifier the voltageproportional to the known term of the corresponding equation togetherwith the fractional voltages proportional to the terms thereof involvingeach of the varia bles and the fractional voltages proportional to theterms thereof involving the time derivatives of each of the variables,thereby enabling the several output circuits to cause the respectivecontrollable means to make the variable voltages individuallyproportional to the solutions of the equations.

3. Means for the solution of a plurality of simultaneous diirerentialequations in a plurality of variables comprising a plurality of channelseach correspondent to a single one of the equations and to one of thevariables and including a source of voltage proportional to a known termof the corresponding equation, a second source of voltage, means forderiving from the second source a variable voltage, means for derivingfrom the variable voltage fractional voltages proportional to the termsin each equation involving directly the corresponding variables, means-l for differentiating successively with respect to time the variablevoltage to derive differential voltages of orders indicated by theequation and means for deriving from the differential voltagesfractional voltages proportional to the terms in each equation involvingthese time derivatives of the one variable, interconnecting means forcombining in each channel the voltages representing for thecorresponding equation respectively the known terms, the terms involvingdirectly each of the variables and the terms involving the various timederivatives of each of the variables, means in each channel responsiveto the sum of the voltages combined therein and controlling thecorresponding variable voltage deriving means to make the sum of thecombined voltages of the several channels separately zero, therebymaking the variable voltages individually proportional to the values ofthe variables constituting the solution desired.

4. In an apparatus for solving for a plurality of variables a pluralityof simultaneous differential equations each including known terms andterms involving one or more of the variables and. their time derivativesof any order, a plurality of interconnected computing channelsindividually corresponding to the equations; each of channels includinga first source of voltage and means for deriving therefrom voltagesproportional to the known terms of the corresponding equation, a secondsource of voltage and means for deriving therefrom a variable voltage,computing means for deriving from the variable voltage voltagesproportional to the terms of all the equations involving the variablesand the time derivatives thereof, a summing amplifier having an inputand an output circuit and motor means responsive to a voltage in theoutput circult to control the second-named voltage deriving means; meansinterconnecting the several channels to combine in the input circuits ofthe respective amplifiers the voltages proportional to the terms of therespective corresponding equation, thereby producing error voltages inthe output circuits of the amplifiers; and means for connecting themotor means in each channel to the output circuit of the amplifiertherein to reduce the error voltages simultaneously to minimum valueswhereby the several variable voltages are made continuously proportionalto the several solutions.

5. An apparatus for solving simultaneously a plurality of ordinarydifferential equations in a plurality of unknown variables in which theindependent variable may be made proportional to time, comprising aplurality of electrical circuits, means for establishing thereinquantities respectively proportional to the known terms of theequations, controllable means for establishing therein arbitraryquantities to represent the unknown variables, computing means includingdifierentiators to perform the mathematical operation on the known andarbitrary quantities necessary to produce quantities respectivelyproportional to each term of the several equations, a plurality of meansfor combining said terms as indicated by the equations to produce aplurality of error voltages, means for causing each of the errorvoltages to control one of the arbitrary quantities in such a way thatall the error voltages are reduced simultaneously to the minimum valuesrequired to actuate the controllable means, whereby the arbitraryquantities are caused to become and remain continuously the solutions ofthe set of equations.

6. Electrical apparatus for solving systems of diiferential equationscomprising signal circuits and means for establishing therein signalscorresponding to the known terms of the equations, means for generatinginitially arbitrary signals representing the unknown quantities in theequations, computing channels including electrical differentiatorsarranged to perform the mathematical operations which the equationsindicate as to be performed on the known and the unknown quantities toproduce the individual terms of the equations and includinginterconnecting means for feeding the known and the unknown terms intosaid computing channels, means for combining the terms as indicated bythe equations to produce error signals, means for util zing each errorsignal to modify the signal representing one of the unknown quantitiesin such a way that all said error signals are reduced to the minimumrequired to control the signal generating means, whereby the unknownquantities become the desired solutions.

CLARENCE A. LOVELL.

REFERENCES CITED The following references are of record in the file ofthis patent:

UNITED STATES PATENTS Number Name Date 2,454,549 Brown et al. Nov. 23,1948 2,455,974 Brown Dec. 14, 1948 2,459,106 Hardy et a1. Jan. 1, 1949

